What is a single-word adjective to define someone in power by virtue of contacts?

If not a single-word adjective, then anything as concise that may define the following traits an individual possess:

• They possess power, in terms that they can make other people do things at their whim.
  It doesn't matter what the source of their power is: could be influence, ability to convince or confuse, threat, by virtue of age/experience/respect earned.
  For instance, each example from different category: "Godfather" from that movie, politicians, subject matter experts, teacher, police.

• They have the right contacts, each for a specific job. - Essencially a "Network of people at their disposal" (e.g politicians and businessmen).
  They know somebody on whom they possess power (above point), and that individual is either authority, or empowered enough to pull in a favour (or their instruction).

Note: It'll be great to know an adjective that need not necessarily imply a criminal intent. A neutral adjective would be preferred.

An adjective I can think of closely resembling this would be "stature", however it requires context to be clear.

Solution 1:

To borrow from your question, you might consider influential.

: exerting or possessing influence

Similarly, I'd suggest well-connected:

: having important and powerful friends

Definitions from Merriam-Webster.